Introduction

Coordinate systems used in near-Earth space physics have been well covered by the works of Russell (1971) and Hapgood (1992). But there has been a lack of publicly available documentation on coordinate systems used in heliospheric space missions and in many cases the information does not seem comprehensive enough for reference purposes². Specifically descriptions of systems based on the physical ephemeris of the Sun and planets and systems based on spacecraft position are currently not available in a form that makes the relation between both systems easy to understand. Experience shows that this deficiency leads to misunderstandings and errors in the production of spacecraft data sets. Another problem is the lack of information on the precision of transformations. This document tries to collect all information necessary for the calculation of coordinate transformations in space science and determines the precision of these transformations whenever possible.

We base all calculations on the current edition of the Astronomical Almanac (2000), hereafter cited as A.³and the Expl.Suppl. (1992), hereafter cited as S.. This means that the base system of astronomical constants used is the IAU(1976) system described in Astr.Alm.Suppl. (1984) implemented in the numerically integrated ephemeris DE200 (Standish, 1990). In general this paper does not describe methods applicable for spatial resolutions below the level of 1 arcsecond but the reader will be able to find the information necessary to achieve higher precision in the cited sources.

To achieve the highest precision in planetary positions one can either (1) implement the numerically integrated ephemeris DE200 or its more precise sequel DE405 (Standish, 1998a)⁴, (2) implement a polynomial expansion of the ephemeris, for example the VSOP87 model(Bretagnon and Francou, 1988)⁵, which is an expansion of DE200, or (3) extend the formulae given in this paper to higher order in time using the values given by Simon et al. (1994) which are also based on VSOP87. Since the extraction code for DE200 is available in different computer languages, its implementation is easy (see e.g. Heafner (1999)) but the size of the corresponding data files may prevent its inclusion in distributed software. For the implementation of VSOP87 we recommend the book by Meeus (2000). In this paper we include first order mean orbital elements from Simon et al. (1994) and give the resulting precision with respect to DE200. The deviations are on the order of arcseconds while differences between DE200 and DE405 are only a few milliarcseconds.

We should point out that for purposes of spacecraft navigation or problems of planetary encounters it is recommended to install a tested software system whenever this is provided by the respective spacecraft navigation team. For most NASA missions such a system is available in the form of the JPL SPICE system ⁶. The SPICE system is a software library which implements DE200 and other reference systems in the form of position and attitude data files ('SPICE kernel files') for solar system bodies and spacecraft. Unfortunately SPICE kernels do not cover all NASA missions and the precision of reconstructed trajectory data is usually not provided. Detailed documentation on SPICE is only available via software file headers, this paper may provide a useful introduction to the principles implemented in SPICE and similar software packages. Before considering implementing formulae given in this paper in your own software package, you might consider implementing the systems cited above, though these will not contain all the coordinate systems defined in our paper. Most data in this paper have been cross-checked by recopying them from the text into our software and comparing the results with tested data. To ease the software implementation of formulae given in this paper we are providing all data contained in the paper on our website ⁷, and will provide corrections and updates on that site as long as possible. The website also contains orbital plots used to determine the precision of data given in this paper.

We also cite the formulae and methods given by Hapgood (1992) for geocentric systems, which are based on the Astronomical Almanac for Computers (1988) which is no longer updated by the Nautical Almanac Offices. The formulae used by Hapgood (1992) are first order approximations of the third order formulae given in Expl.Suppl. (1961). We show later that they achieve a precision of about $34''$ for the timespan 1950-2050 if precession and nutation are included. For many practical purposes the first order approximation is sufficient, but a geocentric error of $34''$ corresponds to a distance of 230km at the L1 Lagrangian point which might be of importance for relative timings between spacecraft for geocentric systems. To keep the paper as compact as possible we will give formulae for planetary orbits to first order only but will point the reader to the sources for improving the precision. The formulae for nutation and precession are given to a precision of at least $2''$ for the period 1950-2050, which allows a higher accuracy transformation between inertial systems. Numerical values are either given in decimal degree ($^\circ$) or arcseconds ($''$). Throughout this paper we use Eulerian matrix rotations to describe transformations denoted $E(\Omega,\theta,\phi)$ (see Appendix). A concise explanation of many terms and systems used in this paper may be found in section L of the Astronomical Almanac (2000).